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I'm using a grammar checking service. It tells me these kind of projects is wrong and should be changed to these kinds of projects. I'm not sure about the correctness of this suggestion, so I googled it on the Google Books website. You can take a look at the result here and here.

Both searches returned many results, and these are from books, which I assumed to be correct. Could anyone help to explain? Thanks in advance.

marked as duplicate by FumbleFingers, Eddie Kal, Nathan Tuggy, choster, Jeff Zeitlin Dec 18 '18 at 18:19

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As @Jason Bassford says, the simple answer is that 'these kind of' is going to be considered wrong in any context which isn't quoting somebody else's speech. So, for all intents and purposes, your grammar checking service is right. 'These' is plural, so 'kinds of projects' all needs to plural too. Most of the modern examples found in Google books I'd consider the results of imperfect proof-reading.

The exception to the rule may be the more historical books cited in your search results. I believe we may see in some of them something akin to words like 'coin' where what we usually consider a singular can be used as a plural when speaking in a general sense. For example in phrases like "I paid in coin" and "do you have any coin?" The latter phrase I consider unusual in modern English, but not ungrammatical.

So in Cobett's Parliamentary History we see the phrase "and the sovereign prince hath imparted and withdrawn these kind of favours". My belief is (which a better grammarian than me may be able to confirm) that using singular forms as plurals was more common in the past when speaking in general terms. Cobett uses the phrase twice, and a search through early modern sources in Medieval and Early Modern Sources Online (http://sources.tannerritchie.com/) reveals 'these kind of' is used 72 times, compared with just 27 instances of what we would now consider the grammatically correct 'these kinds of'.

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Both of the following can use based on use case:

This kind of projects:

Suppose you have a kind that several projects belong to it.

or

These kinds of projects.

Suppose you have many kinds that several projects belong to them.

This or These refer to kind not project.

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